Question

If truncated octahedron has x corners y square faces & z hexagonal faces then

Answer 1

x=24, y=6, z=8

Answer 2

A truncated octahedron is an Archimedean solid formed by cutting off the 6 vertices of a regular octahedron.

A regular octahedron has:

• 6 vertices
• 12 edges
• 8 triangular faces

When the 6 vertices of the octahedron are truncated, 6 new faces are created. Since 4 edges meet at each vertex of a regular octahedron, the truncation of a vertex by a plane results in a new face with 4 edges. In a uniform truncation, these new faces are squares.
So, the number of square faces is y = 6.

The original 8 triangular faces of the octahedron are modified by the truncation. Each triangular face has 3 vertices. The truncation removes these vertices, cutting off the corners of the triangle. This process transforms each original triangular face into a hexagonal face.
So, the number of hexagonal faces is z = 8.

To find the number of corners (vertices) of the truncated octahedron, consider the edges of the original octahedron. An octahedron has 12 edges. Each edge connects two vertices. When the vertices are truncated, new vertices of the truncated octahedron are formed along the original edges. Specifically, each original edge is cut into three segments. The two outer segments are part of the edges of the newly formed square faces, and the central segment becomes an edge of the truncated octahedron. The points where the cuts occur on the original edges become the vertices of the truncated octahedron. Since each original edge is cut at two points (one near each end), and there are 12 original edges, the total number of new vertices formed is 12 * 2 = 24.
So, the number of corners (vertices) is x = 24.

Thus, for a truncated octahedron: x = number of corners (vertices) = 24
y = number of square faces = 6
z = number of hexagonal faces = 8

The question is incomplete as it does not state what expression involving x, y, and z needs to be calculated. Based on the context of such problems, a common request is to find the value of x+y+z, which would be 24 + 6 + 8 = 38. However, without the explicit expression, we can only provide the values of x, y, and z.